Introduction:
Understanding and identifying hidden patterns in a number series is an essential skill in mathematics. This article will guide you through finding the missing term in the given series: 7, 15, 32, __, 138, 281. We will explore different methods to identify the pattern and find the missing number.

Method 1: Analyzing Differences

In the first method, we will analyze the differences between consecutive terms of the series. This technique helps in identifying the underlying pattern.

The series is: 7, 15, 32, __, 138, 281.

Step 1: Calculate the first differences.

Nth Term First Difference 7 - 15 - 7 8 15 32 - 15 17 32 x - 32 x 138 - x 138 281 - 138 143 281

Step 2: Calculate the second differences.

First Differences Second Difference 8 - 17 - 8 9 17 26 - 17 9 26

Based on the second differences being constant (9), we can find the missing term as follows:

17 9 26 (for x - 32)

26 9 35 (for 138 - x)

Thus, the missing number in the series is 58.

The complete series is: 7, 15, 32, 58, 138, 281.

Method 2: Doubling Method

In the second method, we will assume the series doubles each time. Let's calculate the terms:

7 × 2 14 14 × 2 28 28 × 2 56 56 × 2 112 112 × 2 224

Thus, the missing number in the series is 112.

Method 3: Recursive Pattern

Let's explore the recursive pattern where each term is derived from the previous term by multiplying by a constant.

7 × 2 14

14 × 2 28

28 × 2 56

56 × 2 112 (missing term)

112 × 2 224

Therefore, the missing number in the series is 112.

Conclusion

In this article, we analyzed three different methods to find the missing term in the series: 7, 15, 32, __, 138, 281. The various methods provided not only confirm that the missing term is 112, but also offer insights into understanding the underlying patterns. This skill is valuable for solving similar problems and understanding number sequences in mathematics.